I.

1. x=1/2, x=2

2. x=2/3, x=-4

3. The interval [-1, 3)

4. x=pi/2, x=3pi/2

5. x=pi/4, x=3pi/4

1. The graph of 2x+4y=8 is a straight line which passes through the points (0,2) and (4,0). (Note that, whenever possible, all intercepts of a graph should be shown.)

2. The graph of f(x)=x²-6x+8 is a parabola with vertex (3,-1) and x-intercepts (2,0) and (4,0). (The vertex can be found either by completing the square in x or by noting that x=3 is halfway between the intercepts, and thus is the location of the vertex.) If space permits, the point (0,8) should be plotted as well.

3. By completing the square in x and in y, the given equation becomes:

(x+3)² + (y-4)² = 25.

Therefore, the center of the circle is (-3,4), and the radius is 5.

4. The line 2x+y=0 has slope -2, so you want the equation of the line with slope -2 which also passes through (-2,2). The solution is y=-2x-2. (This can also be written as 2x+y=-2.)

5. The perimeter of the given triangle is 10+4*5^1/2, since two sides have length 5 and the other has length 80^1/2, which simplifies to 4*5^1/2. The area of the triangle is 10.

1. f(x)=0 ---> dom(f)=(-oo, oo). (Note: since the symbol for infinity isn't on the keyboard, "oo" will have to do.)

2. g(x)=(1-x²)^1/2 ---> dom(g)=[-1,1]

3. h(x)=1/g(x) ---> dom(h) = (-1,1). (Note that the only difference between the domains of g and h is that the quantity 1-x² may be zero in the first, but not in the second.)

1.
g(f(x))=x²-4x+1,
f(g(x))=(x+1)²-4(x+1) = x²-2x-3

2.
f(x+h)=(x+h)²-4(x+h)=x²+2xh+h²-4x-4h,
g(x+h)=x+h+1

Note that, while f(x+h) doesn't seem that "simplified" the way it's written, you should always expand a quantity such as (x+h)² - 4(x+h) in order to see if any like terms will cancel out. None cancelled at this point, but they will in #3 (below)...

3.
[f(x+h)-f(x)]/h simplifies to 2x+h-4,
[g(x+h)-g(x)]/h simplifies to 1

The key to the first of these is to expand f(x+h) in #2; this gives us:

(x²+2xh+h²-4x-4h)-(x²+4x),

which, after cancelling like terms, leaves us with 2xh+h²-4h.

1. The ratio, it turns out, is the solution to the equation x(x+1)=1. Therefore, the solution is x=-(1+5^1/2))/2, or approximately 0.618.

This ratio is known as The Golden Ratio. It occurs often in nature, and it also has been of interest to artists and architects ever since the days of Ancient Greece. The link (click on "The Golden Ratio" in the previous sentence) will take you to a web page with more information about the Golden Ratio.

2. The next 4 perfect numbers are 28, 496, 8128 and 2096128. Needless to say, no one found more than one -- if anyone had come up with 496, I'd have been very impressed! Note that these numbers are not as random as they appear -- look at the prime factorization of each, and you may see the pattern.

For more on perfect numbers, check out The Perfect Number Journey. It's the best introduction to perfect numbers I've seen on the web.

3. The actual proof of this identity (which is usually called "The Fundamental Identity of Trigonometry," or some similar phrase with the words Fundamental and Trigonometry in it) comes from the fact that the sine and cosine functions are defined by points on the unit circle. To review:

Draw a unit circle, and start at the point (1,0). Given a number t, travel around the circle t units in the counterclockwise direction. (For example, if t=2*pi, you will go around the circle exactly once, since its cifcumference is 2*pi.) You will stop at some point in the plane, with coordinates (x,y). Now, we define: cos(t)=x, sin(t)=y.

The Fundamental Identity is found by simply noting that the equation of the unit circle is x²+y²=1. Substitute cos(t) for x and sin(t) for y, and you're done.